# Geometric Langlands conjecture

- kind: problem
- status: solved (evidence tier V3)
- year: 1980
- authors: Drinfeld and Laumon

**Statement.** The categorical geometric Langlands equivalence holds for curves over the complex numbers.

The proof runs across five papers and roughly 800 pages. It settles the geometric analogue of the Langlands correspondence, in which functions on a curve replace numbers. It does not settle the arithmetic Langlands conjectures, but it gives a precise target and a set of tools for them.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 1980s, Drinfeld and Laumon. Closed 2024, Dennis Gaitsgory, Sam Raskin and collaborators.

## Sources

- [Wikipedia: Geometric Langlands correspondence](https://en.wikipedia.org/wiki/Geometric_Langlands_correspondence)

[The Langlands program](/wiki/Space:langlands-program.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations to this page

- related to [langlands-reciprocity](/wiki/Problem:langlands-reciprocity.md) — The original curation links this problem to Geometric Langlands conjecture; the argument for the link has not been written yet.

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